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3h
answered What is a finitary proof?
3h
comment Riemannian metric and geodesic completeness
In dimension 1 there are only two connected complete Riemannian manifolds, namely the circle and the line. If the length is finite it can't be the line. For topological reasons it can't be the circle.
4h
comment Bijection between measurable sets
Sure, write down your favorite definition and then notice that is transforms nicely under scaling (for example, length of open intervals).
4h
comment Cardinality of the set of multiples of “n”
Well the intersection is empty...
4h
answered Bijection between measurable sets
4h
comment Riemannian metric and geodesic completeness
No, "arclength" means with respect to the Riemannian metric. Here by definition time equals distance.
4h
comment Riemannian metric and geodesic completeness
But you don't need to find the arclength parameter. The length of the curve can be calculated with respect to any parameter.
4h
answered Ricci Tensor, Curvature and Scalar Curvature computation from definition
5h
comment Riemannian metric and geodesic completeness
That looks fine. If the curve is parametrized by arclength then the curve having finite length is the same as running off to infinity in finite time.
5h
comment sign determinant $2\times 2$
Your figure is wrong. The second vector should be pointing upward.
5h
comment Is $\mathbb{R}P^n$ “two-sided” in $\mathbb{R}P^{n+1}$
It's not the 2-cell that's wrapped but rather its boundary. If you remove it you are left with an open 2-cell.
5h
revised Induced map on fundamental groups between surfaces
added 3 characters in body
5h
comment Induced map on fundamental groups between surfaces
You "pinch off" the superfluous handles.
5h
answered Is $\mathbb{R}P^n$ “two-sided” in $\mathbb{R}P^{n+1}$
5h
answered Induced map on fundamental groups between surfaces
6h
comment Induced map on fundamental groups between surfaces
It's not that easy to visualize maps between surfaces of genus 2 or more. One way of generating examples is to look at congruence subgroups in arithmetic groups in SL(2,R) but basically it's a world very different from tori.
6h
comment Riemannian metric and geodesic completeness
Here you don't even need the geodesic equation. Integrate the conformal factor to show that you run off to infinity in finite time.
6h
revised Best mathematical object to describe speed
edited tags
7h
comment The standard role of intuitive numbers in the foundations of mathematics
@OlivierBégassat, let me know if you can count up to the number of elementary particles in the universe (plus 1).
7h
comment Riemannian metric and geodesic completeness
Whether or not it is incomplete depends on whether you run off to infinity in finite time. This is consistent with Hopf-Rinow.